Kan-Lin Hsiung Seung-Jean Kim Stephen Boydboyd/papers/pdf/rgp.pdf98 K.-L. Hsiung et al....

Optim Eng (2008) 9: 95–118DOI 10.1007/s11081-007-9025-z

Tractable approximate robust geometric programming

Kan-Lin Hsiung · Seung-Jean Kim · Stephen Boyd

Received: 24 March 2006 / Accepted: 9 August 2007 / Published online: 4 October 2007© Springer Science+Business Media, LLC 2007

Abstract The optimal solution of a geometric program (GP) can be sensitive to vari-ations in the problem data. Robust geometric programming can systematically allevi-ate the sensitivity problem by explicitly incorporating a model of data uncertainty ina GP and optimizing for the worst-case scenario under this model. However, it is notknown whether a general robust GP can be reformulated as a tractable optimizationproblem that interior-point or other algorithms can efficiently solve. In this paper wepropose an approximation method that seeks a compromise between solution accu-racy and computational efficiency.

The method is based on approximating the robust GP as a robust linear program(LP), by replacing each nonlinear constraint function with a piecewise-linear (PWL)convex approximation. With a polyhedral or ellipsoidal description of the uncertaindata, the resulting robust LP can be formulated as a standard convex optimizationproblem that interior-point methods can solve. The drawback of this basic method isthat the number of terms in the PWL approximations required to obtain an acceptableapproximation error can be very large. To overcome the “curse of dimensionality”that arises in directly approximating the nonlinear constraint functions in the originalrobust GP, we form a conservative approximation of the original robust GP, whichcontains only bivariate constraint functions. We show how to find globally optimalPWL approximations of these bivariate constraint functions.

K.-L. Hsiung (�) · S.-J. Kim · S. BoydInformation Systems Laboratory, Electrical Engineering Department, Stanford University, Stanford,CA 94305-9510, USAe-mail: [email protected]

S.-J. Kime-mail: [email protected]

S. Boyde-mail: [email protected]

96 K.-L. Hsiung et al.

Keywords Geometric programming · Linear programming · Piecewise-linearfunction · Robust geometric programming · Robust linear programming · Robustoptimization

1 Introduction

1.1 Geometric programming

The convex function lse : Rk → R, defined as

lse(z1, . . . , zk) = log(ez1 + · · · + ezk ), (1)

is called the (k-term) log-sum-exp function. (We use the same notation, no matterwhat k is; the context will always unambiguously determine the number of exponen-tial terms.) When k = 1, the log-sum-exp function reduces to the identity.

A geometric program (in convex form) has the form

minimize cT y

subject to lse(Aiy + bi) ≤ 0, i = 1, . . . ,m,

Gy + h = 0,

(2)

where the optimization variable is y ∈ Rn and the problem data are Ai ∈ RKi×n, bi ∈RKi , c ∈ Rn, G ∈ Rl×n, and h ∈ Rl . We call the inequality constraints in the GP (2)log-sum-exp (inequality) constraints. In many applications, GPs arise in posynomialform, and are then transformed by a standard change of coordinates and constraintfunctions to the convex form (2); see Appendix 1. This transformation does not inany way change the problem data, which are the same for the posynomial form andconvex form problems.

Geometric programming has been used in various fields since the late 1960s; earlyapplications of geometric programming can be found in the books Avriel (1980),Duffin et al. (1967), Zener (1971) and the survey papers Ecker (1980), Peterson(1976), Boyd et al. (2007). More recent applications can be found in various fieldsincluding circuit design (Boyd et al. 2005; Chen et al. 2000; Dawson et al. 2001;Daems et al. 2003; Hershenson 2002; Hershenson et al. 2001; Mohan et al. 2000;Sapatnekar 1996; Singh et al. 2005; Sapatnekar et al. 1993; Young et al. 2001), chem-ical process control (Wall et al. 1986), environment quality control (Greenberg 1995),resource allocation in communication systems (Dutta and Rama 1992), informationtheory (Chiang and Boyd 2004; Karlof and Chang 1997), power control of wirelesscommunication networks (Kandukuri and Boyd 2002; O’Neill et al. 2006), queueproportional scheduling in fading broadcast channels (Seong et al. 2006), and statis-tics (Mazumdar and Jefferson 1983).

Algorithms for solving geometric programs appeared in the late 1960s, and re-search on this topic continued until the early 1990s; see, e.g., Avriel et al. (1975),Rajpogal and Bricker (1990). A huge improvement in computational efficiency wasachieved in 1994, when Nesterov and Nemirovsky developed provably efficient

Tractable approximate robust geometric programming 97

interior-point methods for many nonlinear convex optimization problems, includingGPs (Nesterov and Nemirovsky 1994). A bit later, Kortanek, Xu, and Ye developeda primal-dual interior-point method for geometric programming, with efficiency ap-proaching that of interior-point linear programming solvers (Kortanek et al. 1997).

1.2 Robust geometric programming

In robust geometric programming (RGP), we include an explicit model of uncertaintyor variation in the data that defines the GP. We assume that the problem data (Ai, bi)

depend affinely on a vector of uncertain parameters u, that belongs to a set U ⊆ RL:

(Ai(u), bi(u)) =(

A0i +

L∑j=1

ujAji , b

0i +

L∑j=1

ujbji

), u ∈ U ⊆ RL. (3)

The data variation is described by Aji ∈ RKi×n, b

ji ∈ RKi , and the uncertainty set U .

We assume that all of these are known.We consider two types of uncertainty sets. One is polyhedral uncertainty, in

which U is a polyhedron, i.e., the intersection of a finite number of halfspaces:

U = {u ∈ RL | Du � d}, (4)

where d ∈ RK , D ∈ RK×L, and the symbol � denotes the componentwise inequalitybetween two vectors: w � v means wi ≤ vi for all i. The other is ellipsoidal uncer-tainty, in which U is an ellipsoid:

U = {u + Pρ | ‖ρ‖2 ≤ 1, ρ ∈ RL}, (5)

where u ∈ RL and P ∈ RL×L. Here, the matrix P describes the variation in u andcan be singular, in order to model the situation when the variation in u is restrictedto a subspace. Note that due to the affine structure in (3), the ellipsoid uncertainty setU can be transformed to a unit ball (i.e., P can be assumed to be an identity matrix)without loss of generality.

A (worst-case) robust GP (RGP) has the form

minimize cT y

subject to supu∈U

lse(Ai(u)y + bi (u)) ≤ 0, i = 1, . . . ,m,

Gy + h = 0.

(6)

The inequality constraints in the RGP (6) are called robust log-sum-exp (inequality)constraints.

The RGP (6) is a special type of robust convex optimization problem; see, e.g.,Ben-Tal and Nemirovski (1998) for more on robust convex optimization. Unlikethe various types of robust convex optimization problems that have been stud-ied in the literature (e.g., Ben-Tal and Nemirovski 1999; Ben-Tal et al. 2002;Ghaoui and Lebret 1997, 1998; Goldfarb and Iyengar 2003; Boni et al. 2007), the


computational tractability of the RGP (6) is not clear; it is not yet known whether onecan reformulate a general RGP as a tractable optimization problem that interior-pointor other algorithms can efficiently solve.

1.3 Brief overview and outline

We first observe that a log-sum-exp function can be approximated arbitrarily wellby a piecewise-linear (PWL) convex function. Using these approximations, the RGPcan be approximated arbitrarily well as a robust LP, with polyhedral or ellipsoidaldata uncertainty. Since robust LPs, with polyhedral or ellipsoidal uncertainty, can betractably solved (see Appendix 2), this gives us an approximation method for theRGP. In fact, this general approach can be used for any robust convex optimizationproblem with polyhderal or ellipsoidal uncertainty. Piecewise-linear approximationhas been used in prior work on approximation methods for nonlinear convex op-timization problems, since it allows us to approximately solve a nonlinear convexproblem by solving a linear program; see, e.g., Ben-Tal and Nemirovski (2001), Fei-joo and Meyer (1988), Glineur (2000), Thakur (1978).

The problem with the basic PWL approach is that the number of terms needed ina PWL approximation of the log-sum-exp function (1), to obtain a given level of ac-curacy, grows rapidly with the dimension k. Thus, the size of the resulting robust LPis prohibitively large, unless all Ki are small. To overcome this “curse of dimension-ality”, we propose the following approach. We first replace the RGP with a new RGP,in which each log-sum-exp function has only one or two terms. This transformationto a two-term GP is exact for a nonrobust GP, and conservative for a RGP. We thenuse the PWL approximation method on the reduced RGP.

In Sect. 2, we show how PWL approximation of the constraint functions in theRGP (6) leads to a robust LP. We also describe how to approximate a general RGPwith a more tractable RGP which contains only bivariate constraint functions.

In Sect. 3, we develop a constructive algorithm to solve the best PWL convex lowerand upper approximation problems for the bivariate log-sum-exp function. Some nu-merical examples are presented in Sect. 4. Our conclusions are given in Sect. 5. Sup-plementary material is collected in the appendices.

2 Solving robust GPs via PWL approximation

2.1 Robust LP approximation

Suppose we have PWL lower and upper bounds on the log-sum-exp function in theith constraint of the RGP (6),

maxj=1,...,Ii

{f T

ijy + g

ij} ≤ lse(y) ≤ max

j=1,...,Ji

{f T

ij y + gij }, ∀y ∈ RKi ,


where fij, f ij ∈ RKi and g

ij, gij ∈ R. Replacing the log-sum-exp functions in the

RGP (6) with the PWL bounds above, we obtain the two problems

minimize cT y

subject to supu∈U

maxj=1,...,Ji

{f T

ij Ai(u)y + fT

ij bi(u) + gij } ≤ 0, i = 1, . . . ,m,

Gy + h = 0,

(7)

and

minimize cT y

subject to supu∈U

maxj=1,...,Ii

{f T

ijAi(u)y + f T

ijbi(u) + g

ij} ≤ 0, i = 1, . . . ,m,

Gy + h = 0.

(8)

These problems can be reformulated as the robust LPs

minimize cT y

subject to supu∈U

{f T

ij Ai(u)y + fT

ij bi(u) + gij } ≤ 0, i = 1, . . . ,m, j = 1, . . . , Ji,

Gy + h = 0, (9)

and

minimize cT y

subject to supu∈U

{f T

ijAi(u)y + f T

ijbi(u) + g

ij} ≤ 0, i = 1, . . . ,m, j = 1, . . . , Ii ,

Gy + h = 0. (10)

With a polyhedral uncertainty set, these can be cast as (larger) LPs, and for ellipsoidaluncertainty sets, they can be cast as SOCPs; see Appendix 2.

Note that an optimal solution, say y, of the robust LP (9) is also a feasible solutionto the RGP (6). In other words, the robust LP (9) gives a conservative approximationof the RGP (6). The robust LP (10) has the opposite property: its feasible set coversthe feasible set of the RGP (6). Therefore, the optimal value of the robust LP (10),say, cT y, gives a lower bound on the optimal value of the original RGP (6), and inparticular, allows us to bound the error in the feasible, suboptimal point y, for theRGP. In other words, we have

0 ≤ cT (y − y�) ≤ cT (y − y), (11)

where y� is an optimal solution of the RGP. Finally, it is not difficult to see that asthe PWL convex approximations of the log-sum-exp functions are made finer, theoptimal values of the robust LPs (9) and (10) get closer to that of the RGP (6).


2.2 Tractable robust GP approximation

The RGP (6) can be reformulated as another RGP

minimize cT η

subject to supu∈U

lse((ai1 + Bi1u)T η, . . . , (aiKi+ BiKi

u)T η) ≤ 0, i = 1, . . . ,m,

Gη + h = 0 (12)

with the optimization variables η = (y, t) ∈ Rn × R. Here the problem data

c ∈ Rn+1, G ∈ R(l+1)×(n+1), h ∈ Rl+1,

ais ∈ Rn+1, Bis ∈ R(n+1)×L

can be readily obtained from the problem data of the RGP (6); see Appendix 3 forthe details. The RGPs (6) and (12) are equivalent: y ∈ Rn is feasible to (6) if andonly if (y, t) ∈ Rn+1 is feasible to (12) for some t ∈ R. In the following we forma conservative approximation of the RGP (12), in which all the nonlinear constraintfunctions are bivariate.

Consider a k-term robust log-sum-exp constraint in the following generic form:

supu∈U

lse((a1 + B1u)T η, . . . , (ak + Bku)T η) ≤ 0, (13)

where ai ∈ Rn+1, Bi ∈ R(n+1)×L. An approximate reduction procedure, described inAppendix 4, shows that η ∈ Rn+1 satisfies (13) if there exists z = (z1, . . . , zk−2) ∈Rk−2 such that (η, z) satisfies the following system of k − 1 two-term robust log-sum-exp constraints:

supu∈U

lse((a1 + B1u)T η, z1) ≤ 0,

supu∈U

lse((as+1 + Bs+1u)T η − zs, zs+1 − zs) ≤ 0, s = 1, . . . , k − 3, (14)

supu∈U

lse((ak−1 + Bk−1u)T η − zk−2, (ak + Bku)T η − zk−2) ≤ 0,

in which all the constraint functions are bivariate. We will call (14) a “two-term (con-servative) approximation” of the k-term robust log-sum-exp constraint (13).

The idea of tractable RGP approximation is simple: we replace every robust log-sum-exp constraint (with more than two terms) by its two-term conservative approx-imation to obtain a “two-term RGP”, which gives a conservative approximation ofthe original RGP. Although with more variables and constraints, the two-term RGP ismuch more tractable, in the sense that we can approximate the bivariate log-sum-expfunction well with a small number of hyperplanes, as described in Sect. 3. Thenthe two-term RGP can be further solved via robust LP approximation, as shownin Sect. 2.1.

Now we give an exact expression of the two-term RGP approximation. First notethat a one-term robust log-sum-exp constraint is simply a robust linear inequality.


Since no PWL approximation for a one-term constraint is necessary, we can simplykeep all the one-term constraints of a RGP in its two-term RGP approximation (andthe consequent robust LP approximation). Therefore for simplicity, in the followingwe assume all the robust log-sum-exp constraints in RGP (12) have at least two terms,i.e., Ki ≥ 2, i = 1, . . . ,m. The two-term RGP has the form

minimize cT x

subject to supu∈U

lse((a1i + B1

i u)T x, (a2i + B2

i u)T x) ≤ 0, i = 1, . . . ,Kc,

Gx + h = 0,

(15)

where the optimization variables are x = (y, t, z) ∈ Rn × R × RKv , and the problemdata are

aji ∈ Rn+Kv+1, B

ji ∈ R(n+Kv+1)×L, i = 1, . . . ,Kc, j = 1,2,

c = (c,0) ∈ Rn+Kv+1, G = [G 0] ∈ R(l+1)×(n+Kv+1), h = h ∈ Rl+1.

Here Kv = ∑mi=1(Ki −2) is the number of additional variables and Kc = ∑m

i=1(Ki −1) is the number of two-term log-sum-exp constraints.

With general uncertainty structures, the RGP (15) is a conservative approximationof the original RGP (6). In other words, if x = (y, t , z) ∈ Rn × R × RKv is feasibleto (15), y is feasible to (6). Hence the optimal value of the two-term RGP (15), iffeasible, is an upper bound on that of the RGP (6).

3 PWL approximation of two-term log-sum-exp function

There has been growing interest in approximation and interpolation with convexityconstraints (Beatson 1981, 1982; Gao et al. 1995; Hu 1991; McAllister and Roullier1978). However, relatively little attention has been paid to the best PWL convex ap-proximation problem for multivariate, or even bivariate, convex functions. (A heuris-tic method, based on the K-means clustering algorithm, is developed in Magnaniand Boyd (2006).) In this section, the problem of finding the best PWL convex ap-proximation of the two-term (i.e., bivariate) log-sum-exp function is solved and aconstructive algorithm is provided.

3.1 Definitions

Let intX denote the interior of X ⊆ Rm. A function h : Rm → R is called (r-term)piecewise-linear if there exists a partition of Rm as

Rm = X1 ∪ X2 ∪ · · · ∪ Xr,

where intXi �= ∅ and intXi ∩ intXj = ∅ for i �= j , and a family of affine func-tions aT

1 x + b1, . . . , aTr x + br such that h(x) = aT

i x + bi for x ∈ Xi . If an r-termPWL function h is convex, it can be expressed as the maximum of r affine functions:


h(x) = max{aT1 x + b1, . . . , a

Tr x + br}. (See, e.g., Boyd and Vandenberghe (2004).)

Let Pmr denote the set of r-term PWL convex functions from Rm into R. Note that

h ∈ P1r if and only if there exist xi, i = 1, . . . , r − 1 and ai, bi, i = 1, . . . , r with

x1 < · · · < xr−1 and a1 < · · · < ar such that h can be expressed as

h(x) =⎧⎨⎩

a1x + b1, x ∈ (−∞, x1],aix + bi, x ∈ [xi−1, xi], i = 2, . . . , r − 1,

arx + br, x ∈ [xr−1,∞).

The points x1, . . . , xr−1 are called the break points of h, and the affine functionsaix + bi, i = 1, . . . , r are called the segments.

Let f be a continuous function from Rm into R. A function h : Rm → R is calledan r-term PWL convex lower (respectively, upper) approximation to f if h ∈ Pm

r

and h(x) ≤ f (x) (respectively, h(x) ≥ f (x)) for all x ∈ Rm. An r-term PWL convexlower (respectively, upper) approximation f

r∈ Pm

r (respectively, f r ∈ Pmr ) to f

is called a best r-term PWL convex lower (respectively, upper) approximation if ithas the minimum approximation error in the uniform norm among all r-term PWLconvex lower (respectively, upper) approximations to f , which is denoted by εf (r)

(respectively, εf (r)):

εf (r) = supx∈Rm

(f (x) − fr(x))

= infh∈Pm

r

{sup

x∈Rm

(f (x) − h(x))

∣∣∣ h(x) ≤ f (x), ∀x ∈ Rm},

εf (r) = supx∈Rm

(f r(x) − f (x))

= infh∈Pm

r

{sup

x∈Rm

(h(x) − f (x))

∣∣∣ h(x) ≥ f (x), ∀x ∈ Rm}.

3.2 Best PWL approximation of two-term log-sum-exp function

3.2.1 Equivalent univariate best approximation problem

Finding the best r-term PWL convex approximation to the two-term log-sum-expfunction is a “bivariate” best approximation problem over P2

r . In the following weshow that this bivariate best approximation problem can be simplified as an equivalent“univariate” best approximation problem over P1

r .We define the function φ : R → R as

φ(x) = log(1 + ex). (16)

Note that φ satisfies

limx→−∞φ(x) = lim

x→∞(φ(x) − x) = 0. (17)

Thus,

εφ(1) = inf(a,b)∈R2

supx∈R

(φ(x) − ax − b) = ∞, (18)


εφ(2) = supx∈R

(φ(x) − max{0, x}) = log 2. (19)

Now, note that the two-term log-sum-exp function can be expressed as

lse(y1, y2) = y1 + φ(y2 − y1) = y2 + φ(y1 − y2), ∀(y1, y2) ∈ R2. (20)

Therefore it follows from (18–20) that the two-term log-sum-exp function cannot beapproximated by a single affine function with a finite approximation error over R2,but has the unique best two-term PWL convex lower approximation h2 : R2 → Rand upper approximation h2 : R2 → R defined as h2(y1, y2) = max{y1, y2} andh2(y1, y2) = max{y1 + log 2, y2 + log 2} respectively.

From now on, we restrict our discussion to the case r ≥ 3. The following propo-sition establishes the uniqueness and some useful properties of the best r-term PWLconvex lower approximation φ

rto φ for r ≥ 3.

Proposition 1 For r ≥ 3, there exist x1, . . . , xr−1 and (a�i , b

�i ) ∈ R2, i = 1, . . . , r − 2

with

x1 < · · · < xr−1, 0 < a�1 < a�

2 < · · · < a�r−2 < 1, (21)

a�i + a�

r−i−1 = 1, b�i = b�

r−i−1, i = 1, . . . , r − 2, (22)

such that the function φ has the unique best r-term PWL convex lower approxima-tion φ

rdefined as

φr(x) =

⎧⎪⎨⎪⎩

0, x ∈ (−∞, x1],a�

i x + b�i , x ∈ [xi, xi+1], i = 1, . . . , r − 2,

x, x ∈ [xr−1,∞).

(23)

Moreover, there exist x1, . . . , xr−2 ∈ R which satisfy

x1 < x1 < x2 < x2 < · · · < xr−2 < xr−2 < xr−1

such that the segments a�1x + b�

1, . . . , a�r−2x + b�

r−2 are tangent to φ at thepoints x1, . . . , xr−2 respectively. Finally, the maximum approximation error occursonly at the break points of φ

r:

φ(x) − φr(x) < εφ(r), x �∈ {x1, . . . , xr−1},

φ(xi) − φr(xi) = εφ(r), i = 1, . . . , r − 1.

The proof of Proposition 1 is straightforward but lengthy, due to many cases and sub-cases that have to be probed. The reader interested in the complete proof is referredto Hsiung et al. (2006).

As a consequence of Proposition 1 and (20), we have the following corollary.


Corollary 1 For r ≥ 3, the unique best r-term PWL convex lower approxima-tion hr : R2 → R of the two-term log-sum-exp function is

hr(y1, y2) = max{y1, a�r−2y1 + a�

1y2 + b�1, a

�r−3y1 + a�

2y2 + b�2, . . . ,

a�1y1 + a�

r−2y2 + b�r−2, y2} (24)

and the unique best r-term PWL convex upper approximation hr : R2 → R is

hr(y1, y2) = hr(y1, y2) + εφ(r), (25)

where a�i , b

�i , i = 1, . . . , r −2 are the coefficients of the segments of φ

rdefined in (23).

The proof is given in Appendix 5.This corollary shows that both the best r-term PWL convex upper and lower ap-

proximations to the two-term log-sum-exp function can be readily obtained, providedthat φ

ris given. Hence we can restrict our attention on solving the best PWL convex

lower approximation problem for the univariate function φ.

3.2.2 Constructive algorithm

Proposition 1 implies that a function h ∈ P1r (r ≥ 3) with r − 1 break points x1 <

· · · < xr−1 solves the best PWL convex lower approximation problem for φ withapproximation error ε ∈ (0, log 2) (i.e., h ≡ φ

rand ε = εφ(r)) if and only if

h(x) ≤ φ(x), ∀x ∈ R, (26)

limx→−∞h(x) − φ(x) = 0, lim

x→∞h(x) − φ(x) = 0, (27)

h(xi) − φ(xi) = ε, i = 2, . . . , r − 2, (28)

x1 = log(eε − 1), (29)

xr−1 = − log(eε − 1), (30)

and there exist x1, . . . , xr−2 ∈ R such that

h(xi) − φ(xi) = 0, i = 1, . . . , r − 2, (31)

x1 < x1 < x2 < x2 < · · · < xr−2 < xr−2 < xr−1. (32)

Using these properties of the best r-term best PWL convex lower approximation,for any given ε ∈ (0, log 2) and r ≥ 3, the following algorithm can verify if ε = εφ(r)

holds.

given ε ∈ (0, log 2), r ≥ 3define xε = log(eε − 1) and xε = − log(eε − 1)

k := 1, xε1 := xε

repeat

1. find the line y = aεkx +bε

k passing through the point (xεk ,φ(xε

k )−ε) and tangentto the curve y = φ(x) at a point (xε

k , φ(xεk )) with xε

k > xεk


Fig. 1 An illustration of the algorithm which checks if ε = εφ(r) holds for given ε ∈ (0, log 2) and r ≥ 3.In this example we let ε = 0.3 and r = 3. Since xε

2 > xε , we can conclude that εφ(3) < 0.3

2. find xεk+1 > xε

k such that aεkx

εk+1 + bε

k = φ(xεk+1) − ε

3. k := k + 1

until k ≥ r − 1

This algorithm is illustrated in Fig. 1.Now, define an r-term PWL convex function hε : R → R as

hε(x) = max{0, aε1x + bε

1, . . . , aεr−2x + bε

r−2, x}.Note that xε

1 < · · · < xεr−1 and hε satisfy (26–29), and xε

1 < · · · < xεr−2 satisfies (31–

32). Thus hε ≡ φr

if and only if (30) holds, which further implies

ε = εφ(r) ⇐⇒ xεr−1 = xε. (33)

Moreover, (30) implies

ε < εφ(r) ⇐⇒ xεr−1 < xε, (34)

ε > εφ(r) ⇐⇒ xεr−1 > xε. (35)

Observing (33–35), we can see that the following simple bisection algorithm findsεφ(r) and φ

rfor any given r ≥ 3.

given r ≥ 3 and δ > 0ε := 0 and ε := log 2repeat


1. ε := (ε + ε)/22. find the points xε, xε , the segments aε

kx + bεk, k = 1, . . . , r − 1, and the break

points xεk , k = 1, . . . , r − 1 by the algorithm described above

3. if xεr−1 > xε , ε := ε; otherwise, ε := ε

until |xεr−1 − xε | ≤ δ

let εδ = ε and define an r-term PWL convex function φδr: R → R as

φδ

r(x) = max{0, aε

1x + bε1, . . . , a

εr−2x + bε

r−2, x}

Here, it is easy to see that

limδ→0

supx∈R

|φδ

r(x) − φ

r(x)| = 0, lim

δ→0εδ = εφ(r),

i.e., δ > 0 controls the tolerance.

3.2.3 Some approximation results

Table 1 shows the best r-term PWL convex lower approximation to the two-term log-sum-exp function for r = 2, . . . ,5 and the corresponding approximation error εφ(r).As will be shown in Sect. 4, the approximation method described in Sect. 2.2 with thefive-term PWL convex lower approximation provides a quite accurate approximatesolution for the RGP (6). In practical applications we are usually interested in r in therange 5 ≤ r ≤ 10, but we can estimate the error decay rate for large r . Figure 2 showsthe optimal error εφ(r) for 2 ≤ r ≤ 1000. We observe that the curve is almost linearin log-log scale, and using a least-squares fit to the data points (log r, log εφ(r)), r =2, . . . ,1000, we obtain

log εφ(r) ≈ −2.0215 log r + 0.3457.

In normal scale,

εφ(r) ≈ 1.4130

r2.0215≤

√2

r2.

4 Numerical examples

In the following we use some simple RGP numerical examples to demonstrate therobust LP approximation method described in Sect. 2.1. Practical engineering ap-plications, such as power control in lognormal shadowing channels (Hsiung et al.2005) and robust analog/RF circuit design (Yang et al. 2005), have been reported toreveal the effectiveness of the tractable robust GP approximation method proposedin Sect. 2.2.


Table 1 Some best PWL convex lower approximations to the two-term log-sum-exp function

r Approximation Error ε φ(r) Best r-Term PWL Convex Lower Approximation φr

2 0.693 max{ y1, y2 }

3 0.223

max{ y1,

0.500y1 + 0.500y2 + 0.693,

y2 }

4 0.109

max{ y1,

0.271y1 + 0.729y2 + 0.584,

0.729y1 + 0.271y2 + 0.584,

y2 }

5 0.065

max{ y1,

0.167y1 + 0.833y2 + 0.450,

0.500y1 + 0.500y2 + 0.693,

0.833y1 + 0.167y2 + 0.450,

y2 }

Fig. 2 Approximation error εφ(r) vs. the degree of PWL approximation r in log-logscale: r = 2, . . . ,1000


4.1 Two random families

We consider the following RGP, with 500 optimization variables, 500 two-term log-sum-exp inequality constraints, and no equality constraints:

RL: minimize cT y(36)

subject to supu∈U

lse((a1i + B1

i u)T y, (a2i + B2

i u)T y) ≤ 0, i = 1, . . . ,500.

The optimization variable is y ∈ R500, u ∈ RL represents the uncertain problemdata, B1

i and B2i are sparse matrices in R500×L, and

c = 1 ∈ R500, a1i = a2

i = −1 ∈ R500.

Here, 1 is the vector with all entries one. The uncertainty set U ⊆ RL is given by thebox in RL:

U = {u ∈ RL | ‖u‖∞ ≤ 1}, (37)

where ‖u‖∞ denotes the �∞-norm of u.We generated 20 feasible instances, R1

5, . . . ,R205 , of the RGP (36) with L = 5,

by randomly generating the sparse matrices B1i ,B2

i ∈ R500×5, i = 1, . . . ,500 withsparsity density 0.1 and nonzero entries independently uniformly distributed on theinterval [−1,1]. The family {R1

5, . . . ,R205 } is denoted by F5. With L = 20, we also

generated a family F20 of 20 feasible instances, R120, . . . ,R

2020 , in a similar way.

4.2 Approximation results

Before presenting the approximation results for the two random families F5 and F20,we describe the error measure associated with the approximation method describedin this paper.

Suppose the r-term PWL approximation of the two-term log-sum-exp function isused to obtain approximate solutions of the RGP (36). We call r the degree of PWLapproximation, and call the solution yr of the robust LP (7) corresponding to theRGP (36) the r-term upper approximate solution and the solution y

rof the robust

LP (8) the r-term lower approximate solution. Let yr and y� be an r-term upperapproximate solution and an exact optimal solution of the RGP (36) respectively.Then, ecT y�

is the optimal value of the corresponding RGP in posynomial form. Toexpress the difference between ecT y�

and ecT yr , we use the fractional difference inpercentage α, given by

α = 100

(ecT yr

ecT y�− 1

)= 100(ecT (yr−y�) − 1).

We call the value α the r-term PWL approximation error (in percentage) of theRGP (36).


Fig. 3 Approximation results for the random family F5: the degree of PWL approximation r vs. the meanαr (F5) of the r-term PWL approximation errors in log-log scale. The upper solid line is obtained fromlinear least-squares fitting of the data points (log r, logαr (F5)), r = 3,5,7,9 (shown as circles), while thelower one is obtained from linear least-squares fitting of the data points (log r, logαr (F5)), r = 4,6,8,10

We first describe the approximation results for F5. For each r = 3, . . . ,10,we found the r-term upper approximate solutions yr(1), . . . , yr (20) of the ran-domly generated instances R1

5, . . . ,R205 . We also found the exact optimal solu-

tions y�(1), . . . , y�(20) of the instances, by solving the equivalent GPs with 16,000inequality constraints obtained by replicating the inequality constraints for all ver-tices of the uncertainty box U in (37).

Figure 3 shows the degree of PWL approximation r vs. the mean αr(F5) of ther-term PWL approximation errors 100(ecT (yr (i)−y�(i)) − 1), i = 1, . . . ,20, where

αr(F5) = 1

20

20∑i=1

100(ecT (yr (i)−y�(i)) − 1).

This figure shows that, in the region of interest, αr(F5) decreases faster than quadrati-cally with increasing r , since αr(F5), r = 3,5,7,9 decrease faster than quadratically.The variance of the r-term PWL approximation errors 100(ecT (yr (i)−y�(i)) − 1), i =1, . . . ,20 was found to be less than 10−6, regardless of r . The four-term PWL convexupper approximation therefore provides an approximate solution with less than 1%approximation error quite consistently for each of the randomly generated instancesR1

5, . . . ,R205 .

Note that αr(F5) does not decrease monotonically with increasing r . This ismainly because it does not necessarily hold that

r1 ≥ r2 �⇒ hr2(y1, y2) ≥ hr1(y1, y2) ≥ lse(y1, y2), ∀(y1, y2) ∈ R2,


Fig. 4 Approximation results for the random family F20: the degree of PWL approximation r

vs. the upper bound αr (F20) on the mean αr (F20) of the r-term PWL approximation errorsin log-log scale. The solid line is obtained from linear least-squares fitting of the data points(log r, logαr (F20)), r = 3,4, . . . ,10, shown as circles

although

r1 ≥ r2 �⇒sup

(y1,y2)∈R2(hr1(y1, y2) − lse(y1, y2)) < sup

(y1,y2)∈R2(hr2(y1, y2) − lse(y1, y2)),

where hr denotes the best r-term PWL convex upper approximation to the two-termlog-sum-exp function.

We next describe the approximation results for F20. For each r = 3, . . . ,10, wefound the r-term upper approximate solutions yr(1), . . . , yr (20) of the randomly gen-erated instances R1

20, . . . ,R2020 . Replicating the inequality constraints for all the ver-

tices was not possible for the random family F20, since the corresponding uncertaintybox U has approximately 106 vertices. Thus, it is too expensive to find the optimalsolutions y�(1), . . . , y�(20) of the instances R1

20, . . . ,R2020 . Instead, we found the r-

term lower approximate solutions yr(1), . . . , y

r(20) of the instances R1

20, . . . ,R2020 for

each r = 3, . . . ,10.Note from (11) that

0 ≤ ecT (yr (i)−y�(i)) − 1 ≤ ecT (yr (i)−y

r(i)) − 1, i = 1, . . . ,20.

The mean αr(F20) of the r-term approximation errors ecT (yr (i)−y

r(i)) − 1, i =

1, . . . ,20 is therefore an upper bound on the mean αr(F20) of the r-term approxi-


mation errors ecT (yr (i)−y�(i)) − 1, i = 1, . . . ,20:

αr(F20) = 1

20

20∑i=1

100(ecT (yr (i)−y

r(i)) − 1) ≥ αr(F20)

= 1

20

20∑i=1

100(ecT (yr (i)−y�(i)) − 1).

Figure 4 shows the degree of PWL approximation r vs. αr(F20). This figure showsthat, in the region of interest, αr(F20) decreases faster than quadratically with in-

creasing r . The variance of the upper bounds 100(ecT (yr (i)−y

r(i)) − 1), i = 1, . . . ,20

was found to be less than 10−4, regardless of r . The seven-term PWL convex upperapproximation therefore provides an approximate solution with less than 5% approx-imation error consistently for each of the instances R1

20, . . . ,R2020 .

5 Conclusions

We have described an approximation method for a RGP with polyhedral or ellipsoidaluncertainty. The approximation method is based on conservatively approximating theoriginal RGP (6) with a more tractable robust two-term GP in which every nonlinearfunction in the constraints is bivariate. The idea can be extended to a (small) k-termRGP approximation in which every nonlinear function in the constraints has at mostk exponential terms. The extension relies on accurate PWL approximations of k-termlog-sum-exp functions. We are currently working on the extension using the heuristicfor PWL approximation of convex functions developed in Magnani and Boyd (2006).

Acknowledgements The authors are grateful to anonymous reviewers for helpful comments. This mate-rial is based upon work supported by the National Science Foundation under grants #0423905 and (throughOctober 2005) #0140700, by the Air Force Office of Scientific Research under grant #F49620-01-1-0365,by MARCO Focus center for Circuit & System Solutions contract #2003-CT-888, by MIT DARPA con-tract #N00014-05-1-0700, and by the Post-doctoral Fellowship Program of Korea Science and EngineeringFoundation (KOSEF). The authors thank Alessandro Magnani for helpful comments and suggestions.

Appendix 1: Convex formulation of GP

Let Rn++ denote the set of real n-vectors whose components are positive. Letx1, . . . , xn be n real positive variables. A function f : Rn++ → R, defined as

f (x) = d

n∏j=1

xaj

j , (38)

where d > 0 and aj ∈ R, is called a monomial. A sum of monomials, i.e., a functionof the form

f (x) =K∑

k=1

dk

n∏j=1

xajk

j , (39)


where dk > 0 and ajk ∈ R, is called a posynomial (with K terms).An optimization problem of the form

minimize f0(x)

subject to fi(x) ≤ 1, i = 1, . . . ,m,

hi(x) = 1, i = 1, . . . , l,

(40)

where f0, . . . , fm are posynomials and h1, . . . , hl are monomials, is called a geo-metric program in posynomial form. Here, the constraints xi > 0, i = 1, . . . , n areimplicit. The corresponding robust convex optimization problem is called a RGP inposynomial form.

We assume without loss of generality that the objective function f0 is a monomialwhose coefficient is one:

f0(x) =n∏

j=1

xcj

j .

If f0 is not a monomial, we can equivalently reformulate the GP (40) as the followingGP whose objective function is a monomial:

minimize t

subject to f0(x)t−1 ≤ 1,

fi(x) ≤ 1, i = 1, . . . ,m,

hi(x) = 1, i = 1, . . . , l,

where (x, t) ∈ Rn++ × R++ are the optimization variables.GPs in posynomial form are not (in general) convex optimization problems, but

they can be reformulated as convex problems by a change of variables and a trans-formation of the objective and constraint functions. To show this, we define newvariables yi = logxi , and take the logarithm of the posynomial f of x given in (39)to get

f (y) = log(f (ey1, . . . , eyn)) = log

(K∑

i=1

eaTk y+bk

)= lse(aT

1 y + b1, . . . , aTKy + bK),

where ak = (a1k, . . . , ank) ∈ Rn and bk = logdk , i.e., a posynomial becomes a sumof exponentials of affine functions after the change of variables. (Note that if theposynomial f is a monomial, then the transformed function f is an affine function.)This converts the original GP (40) into a GP:

minimize cT y

subject to lse(aTi1y + bi1, . . . , a

TiKi

y + biKi) ≤ 0, i = 1, . . . ,m,

gTi y + hi = 0, i = 1, . . . , l,

(41)


where aij ∈ Rn, i = 1, . . . ,m, j = 1, . . . ,Ki contain the exponents of the posyno-mial inequality constraints, c ∈ Rn contains the exponents of the monomial objectivefunction of the original GP, and gi ∈ Rn, i = 1, . . . , l contain the exponents of themonomial equality constraints of the original GP.

Appendix 2: Robust linear programming

Consider the robust LP

minimize cT x

subject to supu∈U

(ai + Biu)T x + bi ≤ 0, i = 1, . . . ,m,(42)

where the optimization variable is x ∈ Rn, u ∈ RL represents the uncertain problemdata, the set U ⊆ RL describes the uncertainty in u, and c ∈ Rn, ai ∈ Rn, Bi ∈ Rn×L,b ∈ Rm. When the uncertainty set U is given by a bounded polyhedron or an ellipsoid,the robust LP (42) can be cast as a standard convex optimization problem, as shownbelow.

7.1 Polyhedral uncertainty

Let the uncertainty set U be a polyhedron:

U = {u ∈ RL | Du � d},where D ∈ RK×L and d ∈ RK . We assume that U is non-empty and bounded.

Using the duality theorem for linear programming, we can equivalently reformu-late the robust LP (42) as the following LP:

minimize cT x

subject to DT zi = BTi x, i = 1, . . . ,m,

aTi x + dT zi + bi ≤ 0, i = 1, . . . ,m,

zi ≥ 0, i = 1, . . . ,m,

(43)

where the optimization variables are (x, z1, . . . , zm) ∈ Rn × RK × · · · × RK .

7.2 Ellipsoidal uncertainty

Without loss of generality, we assume that the uncertainty set U is a unit ball:

U = {u ∈ RL | ‖u‖2 ≤ 1}.Then, the robust LP (42) can be cast as the second-order cone program

minimize cT x

subject to aTi x + ‖BT

i x‖2 + bi ≤ 0, i = 1, . . . ,m.

See, e.g., Lobo et al. (1998) for details.


Appendix 3: Reformulation of the robust GP

We start with reformulating the RGP (6) as the equivalent RGP

minimize cT

[y

t

]

subject to supu∈U

lse

((A0

i +L∑

j=1

uj Aji

)[y

t

])≤ 0, i = 1, . . . ,m,

G

[y

t

]+ h = 0,

(44)

where (y, t) ∈ Rn × R are the optimization variables, and the problem data are

c = (c,0) ∈ Rn+1, G =[G 00 1

]∈ R(l+1)×(n+1), h =

[h

−1

]∈ Rl+1,

Aji =

[A

ji b

ji

]∈ RKi×(n+1), i = 1, . . . ,m, j = 0,1, . . . ,L.

Denote the sth row of Aji as a

jTis , s = 1, . . . ,Ki , i.e.,

Aji =

⎡⎢⎢⎣

ajT

i1...

ajTiKi

⎤⎥⎥⎦ ∈ RKi×(n+1), i = 1, . . . ,m, j = 0,1, . . . ,L.

Then the RGP (44) can be readily rewritten as the equivalent RGP (12) with theoptimization variables η = (y, t) ∈ Rn × R and the problem data

ais = a0is ∈ Rn+1,

Bis = [a1is · · · aL

is] ∈ R(n+1)×L, s = 1, . . . ,Ki, i = 1, . . . ,m.

Appendix 4: Details of the two-term robust GP approximation

Consider a k-term log-sum-exp constraint:

supu∈U

lse((a1 + B1u)T η, . . . , (ak + Bku)T η) ≤ 0,

where ai ∈ Rn+1, Bi ∈ R(n+1)×L. It is easy to see that

supu∈U

lse((a1 + B1u)T η, . . . , (ak + Bku)T η)

= supu∈U

lse((a1 + B1u)T η, lse((a2 + B2u)T η, . . . , (ak + Bku)T ))

≤ supu∈U

lse((a1 + B1u)T η, sup

u∈Ulse((a2 + B2u)T η, . . . , (ak + Bku)T η)

).


Therefore a sufficient condition for the k-term robust log-sum-exp constraint (13) isthat there exists z1 ∈ R such that

supu∈U

lse((a1 + B1u)T η, z1) ≤ 0, supu∈U

lse((a2 + B2u)T η, . . . , (ak + Bku)T η) ≤ z1.

(45)Similarly, since

supu∈U

lse((a2 + B2u)T η, . . . , (ak + Bku)T η)

≤ supu∈U

lse((a2 + B2u)T η, sup

u∈Ulse((a3 + B3u)T η, . . . , (ak + Bku)T η)

),

a sufficient condition for (45) is that there exist z1, z2 ∈ R such that

supu∈U

lse((a1 + B1u)T η, z1) ≤ 0,

supu∈U

lse((a2 + B2u)T η, z2) ≤ z1,

supu∈U

lse((a3 + B3u)T η, . . . , (ak + Bku)T η) ≤ z2.

Now it is easy to see that η satisfies (13) if there exists z = (z1, . . . , zk−2) ∈ Rk−2

such that (η, z) satisfies the system of k − 1 two-term robust log-sum-exp con-straints:

supu∈U

lse((a1 + B1u)T η, z1) ≤ 0,

supu∈U

lse((as+1 + Bs+1u)T η, zs+1) ≤ zs, s = 1, . . . , k − 3,

supu∈U

lse((ak−1 + Bk−1u)T η, (ak + Bku)T η) ≤ zk−2,

which is obviously equivalent to (14).

Appendix 5: Proof of Corollary 1

The best PWL convex lower approximation problem for the two-term log-sum-expfunction can be formulated as

minimize sup(y1,y2)∈R2

(lse(y1, y2) − max

i=1,...,r{fi1y1 + fi2y2 + gi}

)

subject to lse(y1, y2) ≥ maxi=1,...,r

{fi1y1 + fi2y2 + gi}, ∀(y1, y2) ∈ R2,

(46)

where fi1, fi2, gi ∈ R, i = 1, . . . , r are the optimization variables. Here, notefrom (20) that


lse(y1, y2) − maxi=1,...,r

{fi1y1 + fi2y2 + gi}= y1 + φ(y2 − y1) − max

i=1,...,r{fi1y1 + fi2y2 + gi}

= φ(y2 − y1) − maxi=1,...,r

{(fi1 + fi2 − 1)y1 + fi2(y2 − y1) + gi}.

Obviously, if

sup(y1,y2)∈R2

(lse(y1, y2) − max

i=1,...,r{fi1y1 + fi2y2 + gi}

)< ∞,

then fi1 + fi2 = 1, i = 1, . . . , r . Hence (46) is equivalent to

minimize sup(y1,y2)∈R2

(y1 + φ(y2 − y1) − max

i=1,...,r{fi1y1 + fi2y2 + gi}

)

subject to y1 + φ(y2 − y1) ≥ maxi=1,...,r

{fi1y1 + fi2y2 + gi}, ∀(y1, y2) ∈ R2,

fi1 + fi2 = 1, i = 1, . . . , r.

(47)

This optimization problem is further equivalent to

minimize supx∈R

(φ(x) − maxi=1,...,r

{cix + di})

subject to φ(x) ≥ maxi=1,...,r

{cix + di}, ∀x ∈ R(48)

in which ci, di ∈ R, i = 1, . . . , r are the optimization variables. If c�i , d

�i ∈ R, i =

1, . . . , r solve (48), then f �i1 = 1 − c�

i , f �i2 = c�

i , g�i = d�

i , i = 1, . . . , r solve (47).Conversely, if f �

i1, f�i2, g

�i ∈ R, i = 1, . . . , r solve (47), then c�

i = 1−f �i1 = f �

i2, d�i =

g�i , i = 1, . . . , r solve (48). Moreover, (47) and (48) have the same optimal value.

Hence it is obvious from Proposition 1 that the two-term log-sum-exp function hasthe unique best r-term PWL convex lower approximation hr , given by (24).

We next show that the best r-term PWL convex upper approximation hr to thetwo-term log-sum-exp function can be obtained from (25). To see this, we cast theoptimization problem (46) as

minimize ε

subject to maxi=1,...,r

{fi1y1 + fi2y2 + gi} ≤ lse(y1, y2), ∀(y1, y2) ∈ R2,

lse(y1, y2) ≤ maxi=1,...,r

{fi1y1 + fi2y2 + gi} + ε, ∀(y1, y2) ∈ R2,

(49)

which is obviously equivalent to

minimize ε

subject to maxi=1,...,r

{fi1y1 + fi2y2 + gi} − ε ≤ lse(y1, y2), ∀(y1, y2) ∈ R2,

lse(y1, y2) ≤ maxi=1,...,r

{fi1y1 + fi2y2 + gi}, ∀(y1, y2) ∈ R2.

(50)


If ε, fi1

, fi2

, gi, i = 1, . . . , r solve (49) and ε, f i1, f i2, gi, i = 1, . . . , r solve (50)

respectively, then ε = ε = εφ(r), fi1

= f i1, f i2= f i2, gi = g

i+ ε, i = 1, . . . , r .

Here, note that the best PWL convex upper approximation problem for the two-termlog-sum-exp function can be formulated exactly as (50). Finally, it is easy to seefrom the uniqueness of the best r-term PWL convex lower approximation to φ thatthe two-term log-sum-exp function has the unique best r-term PWL convex upperapproximation hr , given by (25).

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